random variable

[tanudas]

The probability density of a random variable is
f(x)=ax^2e^(−kx) ; k>0,0≤x≤∞
What is the value of a?

I understand that first we'll have to take the integral of the function with limits 0 and ∞ and then substitute them in the integral obtained. This is where I'm stuck because substituting 0 and ∞ yields an undefined number. I need to know where I'm going wrong and how to proceed with it. Thank you !

[Rajat]

Did you try taking limit of the function as x-> infinity ?

[onionknight]

The function isn't quite clear to me. If you could upload a screenshot or image of the question, then I might be able to help you.

[tanudas]

It's from ISI -2008, Q23.

Rajat - No I haven't tried using limits. How would you go about it using limits ?

[Cermank]

You need to integrate by parts. Remember e raised to a negative power of infinity is zero and inverse of e raised to power 0 is 1.

[Rajat]

Hey Tanu, sorry could not revert earlier.
Cermank is right..

integrate by parts, you will reach a point where you will get an expression of the form
Lt x-> infinity f(x)/g(x). Apply L'Hopital's rule. e^-(infinity) is zero as cermank said.

this should give you a = k^3/2

[tanudas]

Thank you guys